
This construction fails to correctly account for the \(n\) -Rényi reflected entropies at \(m=1\)  even away from the phase transition. To see this consider the difference in free energies of the two saddles:
\(\Delta F &= F_{disconn.} - F_{conn.} = \ln \chi \times ( n(m-1) \ell _{mi} - 2 (n-1) \ell _{ew} )\, \\\ell _{mi} &= 2\ln \frac{1-x}{x} \, \qquad \ell _{ew} = \ln \frac{ 1+ \sqrt{1-x}}{\sqrt{x}}\) 

where \(x\)  is the conformal cross ratio of the end points of the boundary intervals and \(x<1/2\)  in the connected phase. Here \(\ell _{mi}\)  is the difference of lengths for the RT surfaces that compute the mutual information in the connected phase. Also \(\ell _{ew}\)  is the length of the cross section. We have approximated the discrete network by a continuum in order to write simple formulas. If we treat these saddles seriously for all \((m,n)\)  then we find the Rényi entropies for \(n>1\)  at \(m=1\)  are dominated (have smaller free energy) by the disconnected phase and thus vanish to leading order in \(\ln \chi \) . Taking the limit \(n \rightarrow 1\)  after setting \(m=1\)  gives 0 for the reflected entropy even in the connected phase. In particular this indicates an order of limits issue that was first pointed out in Ref. [1]}. This answer is not obviously inconsistent, however we will argue that it is nevertheless incorrect.

While the two bulk solutions are legitimate saddles at integer \((m/2,n)\)  and each carries physical significance, it is not obvious that other semi-classical bulk configurations do not dominate over these. In fact as we will see Section 4, there will be a new class of solutions featuring a new group element \(X\in S_{mn}\)  that can dominate the partition function. We will see how this new element interacts with the existing solutions, and we will argue that phases involving \(X\)  are the correct ones for computing the Rényi entropies at \(m=1\) .While this is somewhat similar to the new “replica symmetry breaking” group elements that are important for negativity computations [2]}, [3]}, [4]}, the details differ significantly. These solutions do not have the same order of limits issue above.

At the phase transition there is considerable uncertainty with how to proceed. So far the calculations we discussed were based on some fixed integer replica number \(n,m>1\) . To obtain the reflected entropy we need to analytically continue \(n,m\rightarrow 1\) . While the solutions we obtained above work w independently, we need to include both of them to make a phase transition. However simply summing the two saddles is known to lead to incorrect results [5]}. The simplest issue that arises is from the normalization. Sending \( n \rightarrow 1\)  we expect:
\(1 = \operatorname{tr}\rho _{AA^*} = \operatorname{tr}\left( \operatorname{tr}_{BB^*} \mathinner {|{\sqrt{\rho _{AB}}}\rangle }\mathinner {\langle {\sqrt{\rho _{AB}}}|} \right)\) 
However since we view the two bulk solutions as independent from one another, each of them contributes unity to the trace exactly at the phase transition. Adding those up we obtain a paradoxical result \(\operatorname{tr}\rho _{AA^*} = 2\) . It is clear that we should somehow treat the two seemingly different saddles as limits of a more general class of solution.

Indeed, around the phase transition point we expect there will be other previously sub-leading saddles that will smooth out the sharp transition. A notable example for such effect is the calculation of entanglement entropy for two disjoint regions [6]}, [7]} (See also Ref. [8]}), where a summation over a larger class of group elements called the non-crossing permutations is performed. In our case the problem is harder due to the appearance of the domain wall \(g_A\leftrightarrow g_B\)  in our bulk solution. However when we include a summation over a larger set of saddles, we also expect to find a smoothing out of the reflected entropy phase transition.

